Stochastic methods for modeling molecular-protein interactions are entirely new approaches to the important biological goal of simulating cellular biology in silico. Though great progress has been made in this direction by computational biologists over the past 15 years, the goal of siliconizing cellular molecular interactions still remains remote. More precisely, the current level of model realism has led to a plateau in the prediction accuracy of molecular interactions. This motivates the development of novel dynamic models that incorporate more extensive biological details and can add layers of realism to the simulations. However, such models are computationally daunting, so that it is critical to develop efficient and accurate computational methods. The purpose is to augment molecular-level understanding and simulation of biological interactions. In particular, by exploiting novel developments in stochastic optimi?ation, the investigators shall significantly improve the prediction of interactions by adding a new dimension of realism. However, for many practical cases the stochastic objective function will become a high dimensional, nonGaussian, nonlinear random field that will be computationally very challenging to optimize. This is a hard problem that the investigators plan to address by developing novel Uncertainty Quantification (UQ) mathematical theory. The specific aims are to: (i) Develop a compact dynamic surrogate model of the stochastic objective function that incorporates the molecular structure uncertainty and molecular properties such as the electrostatic fields by solving the nonlinear Poisson Boltzmann (PB) equation. The stochastic optimization is solved efficiently with a surrogate model. (ii) Analyze the complex analytic regularity properties of the solution of the nonlinear Poisson-Boltzmann equation (and the other molecular properties) with respect to the probabilistic molecular conformation model. (iii) Develop convergence rates of the surrogate model from the complex analytic regularity with respect to the number of realizations of the protein structure (computational complexity). Most protein interactions models based on molecular. structure assume a rigid shape thus leading to erroneous predictions. The investigators propose to significantly improve the prediction of protein interactions by incorporating dynamic uncertainty of the molecular conformational shape. The theory and application of UQ to protein interactions is at its infancy.